Find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)

Step 1 ：To find the inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we can use the formula \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \)

Step 2 ：Substitute the values into the formula, we get \( A^{-1} = \frac{1}{(1)(4)-(2)(3)} \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} \)

Step 3 ：Simplify the fraction to get \( A^{-1} = -2 \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} \)

Step 4 ：Then distribute the -2 to each element in the matrix to get \( A^{-1} = \begin{bmatrix} -8 & 4 \ 6 & -2 \end{bmatrix} \)